A129814 -id:A129814 - cp's OEIS frontend

A129825 a(n) = n!*Bernoulli(n-1), n > 2; a(0)=0, a(1)=1, a(2)=1.

0, 1, 1, 1, 0, -4, 0, 120, 0, -12096, 0, 3024000, 0, -1576143360, 0, 1525620096000, 0, -2522591034163200, 0, 6686974460694528000, 0, -27033456071346536448000, 0, 160078872315904478576640000, 0, -1342964491649083924630732800000, 0, 15522270327163593186886877184000000, 0

Offset: 0

Paul Curtz, Jun 03 2007

sign

Define "conjugated" Bernoulli numbers G(n) via G(0)=0, G(1)=B(0)=1, G(2)=-B(1)=1/2, G(n+1)=B(n), where B(n)=A027641(n)/A027642(n).
The sequence is then defined by a(n) = n!*G(n).
The first differences are 1, 0, 0, -1, -4, 4, 120, -120, -12096, ...
The 2nd differences are -1, 0, -1, -3, 8, 116, -240, -11976, 24192, 3011904, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A001332, A027641, A027642, A129716, A129814, A129826.
Equals second left hand column of A161739 (RSEG2 triangle).
Other left hand columns are A161742 and A161743.
Cf. A094310 [T(n,k) = n!/k], A008277 [S2(n,k); Stirling numbers of the second kind], A028246 [Worpitzky's triangle] and A008955 [CFN triangle].

[n le 2 select Floor((n+1)/2) else Factorial(n)*Bernoulli(n-1): n in [0..40]]; // G. C. Greubel, Apr 26 2024

A129825 := proc(n) if n <= 1 then n; elif n = 2 then 1; else n!*bernoulli(n-1) ; fi; end: # R. J. Mathar, May 21 2009

a[n_] := n!*BernoulliB[n-1]; a[0]=0; a[2]=1; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Mar 04 2013 *)

[(n+1)//2 if n <3 else factorial(n)*bernoulli(n-1) for n in range(41)] # G. C. Greubel, Apr 26 2024

From Johannes W. Meijer, Jun 18 2009: (Start)
a(n) = Sum_{k=1..n} (-1)^(k+1)*(n!/k)*S2(n, k)*(k-1)!.
a(n) = Sum_{k=0..n-1} ((-1)^k/(k!*(k+1)!))*n!*A028246(n, k+1) *A008955(k, k). (End)
a(n) = A129814(n-1) for n > 2. - Georg Fischer, Oct 07 2018

Edited by R. J. Mathar, May 21 2009

A001332 a(n) = Bernoulli(2n) (2*n + 1)!.

Original entry on oeis.org

1, 1, -4, 120, -12096, 3024000, -1576143360, 1525620096000, -2522591034163200, 6686974460694528000, -27033456071346536448000, 160078872315904478576640000, -1342964491649083924630732800000, 15522270327163593186886877184000000

Offset: 0

N. J. A. Sloane

G. S. Kazandzidis, On a Matrix and a Class of Polynomials, Bulletin de la Société Mathématique de Grèce, Nouvelle Série - Vol. 6 I, Fasc. 1, (1965), pp. 105-126.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..50
Zentralblatt, Review of Kazandzidis, On a Matrix and a Class of Polynomials.
Index entries for sequences related to Bernoulli numbers.

Cf. A129814.

Table[BernoulliB[2*n]*(2*n + 1)!, {n, 0, 20}] (* T. D. Noe, Jun 28 2012 *)

{a(n) = if( n<0, 0, (2*n + 1)! * bernfrac( 2*n))} /* Michael Somos, Oct 08 2003 */

Lacunary e.g.f: x / (exp(x) - 1) + x / 2 = Sum_{k>=0} a(k) * x^(2*k) / ((2*k)! * (2*k + 1)!). - Michael Somos, Mar 29 2011
a(n) = determinant of the 2n X 2n matrix ( d(i,j) = binomial( i+1, i-j+2) if j < i+2 else 0 ). - Michael Somos, Oct 08 2003
a(n) = A129814(2*n). - Michael Somos, Mar 29 2011

A191578 Triangle read by rows, based on expansion of (x^2/(exp(x)-1))^m = x^m+sum(n>m T(n,m)m!/((n-m)!n!)*x^n).

Original entry on oeis.org

1, -1, 1, 1, -3, 1, 0, 10, -6, 1, -4, -30, 40, -10, 1, 0, 36, -270, 110, -15, 1, 120, 420, 1596, -1260, 245, -21, 1, 0, -2400, -5040, 14056, -4200, 476, -28, 1, -12096, -30240, -46080, -136080, 72576, -11340, 840, -36, 1, 0, 423360, 756000, 795600, -1197000, 276192, -26460, 1380, -45, 1, 3024000, 5987520, 4213440, 6098400, 17087400, -6652800, 857472, -55440, 2145, -55, 1, 0, -163296000, -251475840, -220651200, -158004000, 151169040, -27941760, 2297592, -106920, 3190, -66, 1

Offset: 1

Vladimir Kruchinin, Jun 07 2011

Expansion of (x*Bernoulli(x)^m=x^m+sum(n>m m!*sum(k=1..n-m, (k!*stirling1(m+k,m)*stirling2(n-m,k))/(m+k)!))/(n-m)!*x^n)
Riordan Array (1,x*Bernoulli(x)) without first column.
Riordan Array (Bernoulli(x),x*Bernoulli(x)) numbering triangle (0,0).

			1,
-1,1,
1,-3,1,
0,10,-6,1,
-4,-30,40,-10,1,
0,36,-270,110,-15,1,
120,420,1596,-1260,245,-21,1

Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
Dmitry V. Kruchinin and Vladimir V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.

First column T(n,1)=A129814(n-1)

A191578 := proc(n, m)
    if m=n then
        1;
    else
        add(combinat[stirling2] (n-m, k) *k! *combinat[stirling1](m+k, m)/(m+k)!, k=1..n-m) ;
        %*n! ;
    end if;
end proc: # R. J. Mathar, Jun 14 2013

t[n_, m_] := n!*Sum[ (k!*StirlingS1[m+k, m]*StirlingS2[n-m, k])/(m+k)!, {k, 1, n-m}]; t[n_, n_] = 1; Table[t[n, m], {n, 1, 12}, {m, 1, n}] // Flatten (* Jean-François Alcover, Feb 22 2013 *)

T(n,m):=n!*sum((k!*stirling1(m+k,m)*stirling2(n-m,k))/(m+k)!,k,0,n-m); /* Vladimir Kruchinin, Jun 14 2013 */

T(n,m):=n!*(n-m)!/m!*sum(k!*binomial(m+k-1,m-1)*sum(((-1)^j*stirling2(n-m+j,j))/((k-j)!*(n-m+j)!),j,0,k),k,0,n-m); /* Vladimir Kruchinin, Jun 14 2013 */

T(n,m)=n!*sum(k=0..n-m, (k!*stirling1(m+k,m)*stirling2(n-m,k))/(m+k)!).

T(n,m):=n!*(n-m)!/m!*sum(k=0..n-m, k!*binomial(m+k-1,m-1)*sum(j=0..k, ((-1)^j*stirling2(n-m+j,j))/((k-j)!*(n-m+j)!))). [Vladimir Kruchinin, Jun 14 2013 ]

A273198 a(n) = T(n,2) with T(n, m) = (mn+1)! Sum_{k=0..n}( 1/(mk+1) Sum_{j=0..mk} (-1)^jC(k,j)j^(mn) ).

Original entry on oeis.org

1, -2, 296, -327984, 1363872384, -15198541159680, 372495898187043840, -17616182020373076940800, 1464370216956293433318604800, -199499758936277018742988067635200, 42181903584776412718275835664105472000, -13251216132203374725100642797337549799424000

Offset: 0

Peter Luschny, Jun 26 2016

sign

G. C. Greubel, Table of n, a(n) for n = 0..123

Cf. A129814, A273196, A273197.

Flatten[{1, Table[(2*n + 1)! * Sum[1/(2*k + 1)*Sum[(-1)^j*Binomial[k, j]*j^(2*n), {j, 0, 2*k}], {k, 0, n}], {n, 1, 10}]}] (* Vaclav Kotesovec, Jun 26 2016 *)

def T(n, m): return factorial(m*n+1) * sum(1/(m*k+1)*sum((-1)^j*binomial(k,j)* j^(m*n) for j in (0..m*k)) for k in (0..n))
def a(n): return T(n, 2)
print([a(n) for n in (0..12)])

a(n) ~ (-1)^n * sqrt(Pi) * 2^(4*n) * n^(4*n + 1/2) / (sqrt(1-c) * exp(4*n) * c^n * (2-c)^(n-1)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... . - Vaclav Kotesovec, Jun 26 2016

A356545 Triangle read by rows. T(n, k) are the coefficients of polynomials p_n(x) based on the Eulerian numbers of first order representing the Bernoulli numbers as B_n = p_n(1) / (n + 1)!.

Original entry on oeis.org

1, 1, 0, 2, -1, 0, 6, -8, 2, 0, 24, -66, 44, -6, 0, 120, -624, 792, -312, 24, 0, 720, -6840, 14496, -10872, 2736, -120, 0, 5040, -86400, 285840, -347904, 171504, -28800, 720, 0, 40320, -1244880, 6181920, -11245680, 8996544, -3090960, 355680, -5040, 0

Offset: 0

Peter Luschny, Aug 11 2022

The Bernoulli numbers with B(1) = 1/2 can be represented as the weighted sum of Eulerian numbers, where we use the definition as given by Graham et al., Eulerian(n, k) = A173018(n, k). For n >= 0 we have
   B_(n) = (1/(n + 1)) * Sum_{k=0..n} (-1)^k * Eulerian(n, k) / binomial(n, k).
The formula was given by Worpitsky in 1883 (see link) as an example for the application of a formula of Schlömilch from 1856. In 2019 the identity was proved in the modern fashion by Gessel on MathOverflow.
For a variant of this identity see the first formula in A356546.
An analogous representation based on the Eulerian numbers of second order is given in A356547.

			The table T(n, k) of the coefficients, sorted in ascending order, starts:
[0]     1;
[1]     1,        0;
[2]     2,       -1,       0;
[3]     6,       -8,       2,         0;
[4]    24,      -66,      44,        -6,       0;
[5]   120,     -624,     792,      -312,      24,        0;
[6]   720,    -6840,   14496,    -10872,    2736,     -120,      0;
[7]  5040,   -86400,  285840,   -347904,  171504,   -28800,    720,     0;
[8] 40320, -1244880, 6181920, -11245680, 8996544, -3090960, 355680, -5040, 0;

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 268. (Since the thirty-fourth printing, Jan. 2022, with B(1) = 1/2.)

Ira Gessel, Eulerian number identity, MathOverflow, Apr 2019.
Peter Luschny, How are the Eulerian numbers of the first-order related to the Eulerian numbers of the second-order?, MathOverflow, Feb. 2021.
Peter Luschny, Eulerian polynomials.
Oskar Schlömilch, Ueber die Bernoulli'sche Funktion und deren Gebrauch bei der Entwickelung halbconvergenter Reihen, Zeitschrift fuer Mathematik und Pysik, vol. 1 (1856), p. 193-211.
Julius Worpitsky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik (Crelle), 94 (1883), 203-232. See page 22, first formula.

Cf. A173018 (Eulerian number), A164555(n)/A027642(n) (Bernoulli numbers with B(1) = 1/2), A129814 (row sums, but different sign for n = 1).

Cf. A098361, A123125, A356546, A356547, A356601, A356602.

E1 := proc(n, k) combinat:-eulerian1(n, k) end:
p := (n, x) -> add(E1(n, k)*k!*(n - k)!*(-x)^k, k = 0..n):
seq(print(seq(coeff(p(n, x), x, k), k=0..n)), n = 0..8);
seq(p(n, 1)/(n + 1)!, n = 0..14); # check the Bernoulli representation

T[n_, k_] := k! * (n-k)! * Sum[(-1)^(k-j) * (k-j+1)^n * Binomial[n+1, j], {j, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // TableForm
(* Diagonals: *)
d[n_, k_] := k! * (n - k)! * Sum[(-1)^(n-k-j)*(n - j - k + 1)^n * Binomial[n + 1, j], {j, 0, n - k}];

Let p_n(x) = Sum_{k=0..n} Eulerian(n, k)*k!*(n - k)! * (-x)^k. For x = 1 these polynomials give rise to the representation Bernoulli(n) = p_n(1) / (n + 1)!.
T(n, k) = [x^k] p_n(x).
T(n, k) = (-1)^k*Eulerian(n, k)*k!*(n - k)!.
T(n, k) = k! * (n-k)! * Sum_{j=0..k} (-1)^(k-j)*(k-j+1)^n*binomial(n+1, j).
T(n, k) = (-1)^k * A173018(n, k) * A098361(n, k).
T(n, k) = (-1)^k * A123125(n, n - k) * A098361(n, n - k).

A137777 Triangular sequence of coefficients from the expansion of the derivative of the Bernoulli polynomial function: p(x,t) = texp(xt)/(exp(t)-1); q(x,t) = p'(x,t) = dp(x,t)/dt.

Original entry on oeis.org

2, -2, 4, 2, -12, 12, 0, 24, -72, 48, -8, 0, 240, -480, 240, 0, -240, 0, 2400, -3600, 1440, 240, 0, -5040, 0, 25200, -30240, 10080, 0, 13440, 0, -94080, 0, 282240, -282240, 80640, -24192, 0, 483840, 0, -1693440, 0, 3386880, -2903040, 725760, 0, -2177280, 0, 14515200, 0, -30481920, 0, 43545600, -32659200

Offset: 0

Roger L. Bagula and Gary W. Adamson_, Apr 28 2008

Row sums are {2, 2, 0, -8, 0, 240, 0, -24192, 0, 6048000, 0, ...}.
From Peter Luschny, Apr 23 2009: (Start)
The sequence can also be computed as the coefficients of the Bernoulli polynomials B_n(x) times 2(n+1)! for n >= 1. As Peter Pein observed the Mathematica code then reduces to
Table[CoefficientList[2 (n+1)! BernoulliB[n,x],x],{n,1,10}] // Flatten
Note that this formula is also well defined in the case n = 0 and has the value 2. (End)

			{2},
{-2, 4},
{2, -12, 12},
{0,24, -72, 48},
{-8, 0, 240, -480, 240},
{0, -240, 0, 2400, -3600, 1440},
{240, 0, -5040, 0, 25200, -30240, 10080},
{0, 13440, 0, -94080, 0, 282240, -282240, 80640},
{-24192, 0, 483840, 0, -1693440, 0, 3386880, -2903040, 725760},
{0, -2177280, 0, 14515200, 0, -30481920, 0, 43545600, -32659200, 7257600},
{6048000, 0, -119750400, 0, 399168000, 0, -558835200, 0, 598752000, -399168000, 79833600},
{0, 798336000, 0, -5269017600, 0, 10538035200, 0, -10538035200, 0, 8781696000, -5269017600, 958003200}

seq(seq(coeff(bernoulli(k,x)*2*(k+1)!,x,i),i=0..k),k=1..10); # Peter Luschny, Apr 23 2009

Clear[p, b, a]; p[t_] = D[t^2*Exp[x*t]/(Exp[t]-1),{t,1}];
a = Table[CoefficientList[2*n!^2*SeriesCoefficient
[Series[p[t],{t,0,30}],n],x],{n,0,10}]; Flatten[a]
Table[CoefficientList[2 BernoulliB[k,x] Gamma[2+k],x],{k,0,10}]//Flatten

p(x,t) = t*exp(x*t)/(exp(t)-1); q(x,t) = p'(x,t) = dp(x,t)/dt = Sum_{n>=0} Q(x,n)*t^n/n!; out_n,m=2*(n + 2)!*n!*Coefficients(Q(x,n).
A137777(n,0) = 2*A129814(n) for n >= 0.
A137777(n,n) = 2*(n+1)! for n >= 0.
Conjecture on row sums: Sum_{k=0..n+1} T(n,k) = 2*A129825(n+2). - R. J. Mathar, Jun 03 2009

Edited by N. J. A. Sloane, Jan 03 2010, incorporating comments from Peter Luschny and Peter Pein

A273196 a(n) = numerator of T(n, 2) with T(n, m) = Sum_{k=0..n}( 1/(mk+1) Sum_{j=0..mk} (-1)^jC(k,j)j^(mn) ).

Original entry on oeis.org

1, -1, 37, -6833, 56377, -439772603, 27217772209, -202070742359, 80837575181815013, -155957202651688954367, 1770963292969902374951, -16092436217742770647634507, 2975968726866580246152132993, -963399772945511487665759472653, 3891037048609240492066339458106680163

Offset: 0

Peter Luschny, Jun 26 2016

T(n,0) are the natural numbers, T(n,1) the Bernoulli numbers.

Cf. A273197 (denominator), T(n,0) = A000027, T(n,1) = A027641/A027642.

Also T(n,1)*(1*n+1)! = A129814, T(n,2)*(2*n+1)! = A273198.

Table[Function[{n, m}, If[n == 0, 1, Numerator@ Sum[1/(m k + 1) Sum[(-1)^j Binomial[k, j] j^(m n), {j, 0, m k}], {k, 0, n}]]][n, 2], {n, 0, 14}] (* Michael De Vlieger, Jun 26 2016 *)

def T(n, m): return sum(1/(m*k+1)*sum((-1)^j*binomial(k,j)*j^(m*n) for j in (0..m*k)) for k in (0..n))
def a(n): return T(n, 2).numerator()
print([a(n) for n in (0..14)])

A273197 a(n) = denominator of T(n, 2) with T(n, m) = Sum_{k=0..n}( 1/(mk+1) Sum_{j=0..mk} (-1)^jC(k,j)j^(mn) ).

Original entry on oeis.org

1, 3, 15, 105, 15, 1155, 455, 15, 19635, 95095, 2145, 31395, 7735, 2805, 10818885, 50115065, 3315, 596505, 80925845, 3795, 18515805, 221847535, 2211105, 204920500785, 1453336885, 148335, 95055765, 287558635, 27897511785, 397299047145, 5613813089885, 8897205

Offset: 0

Peter Luschny, Jun 26 2016

T(n,0) are the natural numbers, T(n,1) the Bernoulli numbers.

Cf. A273196 (numerators).
T(n,0) = A000027(n) for n>=1.
T(n,1) = A027641(n)/A027642(n) for all n>=0.
T(n,1)*(1*n+1)! = A129814(n) for all n>=0.
T(n,2)*(2*n+1)! = A273198(n) for all n>=0.

Table[Function[{n, m}, If[n == 0, 1, Denominator@ Sum[1/(m k + 1) Sum[(-1)^j Binomial[k, j] j^(m n), {j, 0, m k}], {k, 0, n}]]][n, 2], {n, 0, 31}] (* Michael De Vlieger, Jun 26 2016 *)

def T(n, m): return sum(1/(m*k+1)*sum((-1)^j*binomial(k,j)*j^(m*n) for j in (0..m*k)) for k in (0..n))
def a(n): return T(n, 2).denominator()
print([a(n) for n in (0..31)])

cp's OEIS Frontend

A129825 a(n) = n!*Bernoulli(n-1), n > 2; a(0)=0, a(1)=1, a(2)=1.

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A001332 a(n) = Bernoulli(2*n) * (2*n + 1)!.

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A191578 Triangle read by rows, based on expansion of (x^2/(exp(x)-1))^m = x^m+sum(n>m T(n,m)*m!/((n-m)!*n!)*x^n).

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A273198 a(n) = T(n,2) with T(n, m) = (m*n+1)! * Sum_{k=0..n}( 1/(m*k+1) * Sum_{j=0..m*k} (-1)^j*C(k,j)*j^(m*n) ).

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A356545 Triangle read by rows. T(n, k) are the coefficients of polynomials p_n(x) based on the Eulerian numbers of first order representing the Bernoulli numbers as B_n = p_n(1) / (n + 1)!.

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A137777 Triangular sequence of coefficients from the expansion of the derivative of the Bernoulli polynomial function: p(x,t) = t*exp(x*t)/(exp(t)-1); q(x,t) = p'(x,t) = dp(x,t)/dt.

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A273196 a(n) = numerator of T(n, 2) with T(n, m) = Sum_{k=0..n}( 1/(m*k+1) * Sum_{j=0..m*k} (-1)^j*C(k,j)*j^(m*n) ).

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A001332 a(n) = Bernoulli(2n) (2*n + 1)!.

A191578 Triangle read by rows, based on expansion of (x^2/(exp(x)-1))^m = x^m+sum(n>m T(n,m)m!/((n-m)!n!)*x^n).

A273198 a(n) = T(n,2) with T(n, m) = (mn+1)! Sum_{k=0..n}( 1/(mk+1) Sum_{j=0..mk} (-1)^jC(k,j)j^(mn) ).

A137777 Triangular sequence of coefficients from the expansion of the derivative of the Bernoulli polynomial function: p(x,t) = texp(xt)/(exp(t)-1); q(x,t) = p'(x,t) = dp(x,t)/dt.

A273196 a(n) = numerator of T(n, 2) with T(n, m) = Sum_{k=0..n}( 1/(mk+1) Sum_{j=0..mk} (-1)^jC(k,j)j^(mn) ).

A273197 a(n) = denominator of T(n, 2) with T(n, m) = Sum_{k=0..n}( 1/(mk+1) Sum_{j=0..mk} (-1)^jC(k,j)j^(mn) ).